Fluid Dynamics — formula sheet

Reynolds Transport Theorem

The rate at which the total amount of \(\phi\)-stuff carried by a chunk of moving material changes has two contributions. First, the stuff already inside the region can intensify or fade in place: that is \(\int_V \partial_t\phi\,dV\), measured as if the region were frozen in space. Second, as the material boundary sweeps through the field it carries \(\phi\) outward wherever \(\mathbf{v}\cdot\hat{\mathbf{n}}>0\) and inward wherever \(\mathbf{v}\cdot\hat{\mathbf{n}}<0\); the net of this is the surface flux \(\oint_{\partial V}\phi\,\mathbf{v}\cdot\hat{\mathbf{n}}\,dA\). The material change equals fixed-frame change plus what the flow drags across the border.

The Continuity Equation

At a fixed point in space, the density rises exactly as fast as mass is carried inward by the flow. The vector \(\mathbf{j}=\rho\mathbf{u}\) is the mass flux (mass per unit area per unit time); its divergence measures the net outflow per unit volume, and any net outflow must be paid for by a local drop \(\partial\rho/\partial t<0\). Expanding the divergence gives the material-derivative form \(\dfrac{\mathrm{D}\rho}{\mathrm{D}t} = -\rho\,\nabla\cdot\mathbf{u}\): following a parcel, its density changes only through the local volumetric expansion rate \(\nabla\cdot\mathbf{u}\).

Existence of the Cauchy Stress Tensor

The force per unit area transmitted across any internal surface is obtained by contracting a single position-dependent object, the stress tensor \(\boldsymbol{\sigma}\), with the surface's outward unit normal. All the orientation dependence of contact forces at a point lives in these nine numbers \(\sigma_{ij}\): the diagonal entries are normal stresses (tension/compression), the off-diagonal entries are shear stresses. Nothing about the surface beyond its normal matters.

Cauchy's Momentum Equation

Mass-times-acceleration per unit volume of a moving fluid parcel equals the net internal surface force per unit volume (the stress divergence) plus the body force per unit volume. The acceleration on the left is the material acceleration \(D\mathbf{u}/Dt\), which includes the convective part \((\mathbf{u}\cdot\nabla)\mathbf{u}\): even in a steady field a parcel accelerates as it moves through spatial gradients. The divergence of the stress measures the imbalance of tractions on opposite faces of an infinitesimal element; a uniform stress exerts no net force.

Navier-Stokes from the Newtonian Constitutive Law

The acceleration of a fluid parcel (local plus advective) is driven by three forces per unit mass: the pressure gradient pushes fluid from high to low pressure; the viscous term \( \nu\nabla^2\vec{v} \) diffuses momentum, smoothing velocity differences exactly as heat conduction smooths temperature; and \( \vec{f} \) is any body force such as gravity. The incompressibility constraint acts as the equation that determines the pressure field, so that the velocity stays divergence-free at every instant.

Euler's Equations as the Inviscid Limit

Bernoulli's Theorem, Steady and Unsteady

Each term is energy per unit mass. The steady relation says that for an ideal fluid the sum of kinetic energy \( \tfrac12 v^2 \), flow work / pressure energy \( p/\rho \), and gravitational potential \( gz \) is conserved as a fluid parcel travels along its streamline: where it speeds up, pressure or height must fall. The unsteady form adds the local term \( \partial\varphi/\partial t \), the reversible work done to accelerate the whole irrotational field, and upgrades "along a streamline" to "at every point simultaneously".

The Vorticity Transport Equation

Following a fluid element, its vorticity changes only through two mechanisms. First, the local velocity gradient acting on the existing vorticity: the component of \(\mathbf{S}\vec{\omega}\) along \(\vec{\omega}\) is stretching (a vortex line pulled taut spins faster, conserving angular momentum), and the component perpendicular to \(\vec{\omega}\) is tilting (a vortex line reoriented by shear). Second, viscous diffusion, which smooths vorticity gradients and ultimately dissipates rotation. Pressure and conservative body forces are absent — they cannot create vorticity in a constant-density fluid.

Kelvin's Circulation Theorem

The velocity circulation around a loop of marked fluid particles never changes, no matter how violently the loop is stretched, twisted, or advected. By Stokes' theorem the same statement is that the vorticity flux through any material surface bounded by the loop is frozen — the geometric content behind "vortex lines move with the fluid" (Helmholtz's theorem).

Potential Flow and Laplace's Equation

Incompressible + irrotational is exactly the kinematic content of "harmonic potential". The velocity is read off as the gradient of a scalar that obeys Laplace's equation; boundary conditions (typically \( \partial\phi/\partial n \) prescribed on solid walls, the no-penetration condition) close the problem. Because Laplace's equation is linear, elementary solutions superpose, and because it is elliptic, \( \phi \) is smooth in the interior and determined everywhere by data on the boundary.

D'Alembert's Paradox

In steady, incompressible, irrotational, inviscid flow the pressure field around any finite closed body is fore–aft balanced: whatever the front pushes back, the rear pushes forward by exactly as much. The high pressure at the forward stagnation point is mirrored by an equal high pressure at the rear stagnation point, and the suction over the shoulders is symmetric. No net force of any kind acts on the body — no drag, and (absent circulation) no lift.

Stokes Drag on a Sphere

The drag on a slowly-moving sphere is proportional to viscosity, radius, and speed — each to the first power. Of the total, two-thirds (\(4\pi\mu aU\)) is skin-friction from the tangential shear and one-third (\(2\pi\mu aU\)) is form drag from the fore–aft pressure difference. Crucially the fluid density does not appear: at \(\mathrm{Re}\to0\) inertia plays no role, so the drag is purely viscous.

The Reynolds Number from Nondimensionalisation

The Reynolds number is the ratio of the characteristic inertial momentum flux \(\rho U^2/L\) to the characteristic viscous stress gradient \(\mu U/L^2\). When \(\mathrm{Re}\gg1\) inertia dominates and viscosity acts only in thin layers; when \(\mathrm{Re}\ll1\) viscosity dominates and inertia is negligible. Two geometrically similar flows with the same \(\mathrm{Re}\) obey the identical dimensionless equation with identical dimensionless boundary conditions, so their nondimensional fields \(\mathbf{u}^*,p^*\) coincide — this is dynamic similarity.

Energy Balance and Viscous Dissipation

The kinetic energy of a material element changes by the power of gravity, the net working of surface stresses (a divergence, so a pure transport that integrates to a boundary flux), a reversible exchange with internal energy through \(p(\nabla\!\cdot\!\mathbf{u})\), and a strictly non-negative loss \(\Phi\). Only \(\Phi\) is one-way: it is the local rate per unit volume at which ordered kinetic energy is degraded into disordered thermal motion. For incompressible flow \(\nabla\!\cdot\!\mathbf{u}=0\) and \(\Phi=2\mu\,S_{ij}S_{ij}\).

The Acoustic Wave Equation and Speed of Sound

Each acoustic perturbation obeys the classical wave equation, so disturbances propagate non-dispersively at a single speed \(c\) set by the isentropic stiffness of the medium, \(\left(\partial p/\partial\rho\right)_s\). Stiffer or springier fluids (large \(\partial p/\partial\rho\)) carry sound faster; denser fluids at fixed stiffness carry it slower, because \(\partial p/\partial\rho=(\partial p/\partial\rho)\) already folds in the inertia \(\rho\). For an ideal gas \(c=\sqrt{\gamma p_0/\rho_0}=\sqrt{\gamma R T/M}\).

Prandtl's Boundary-Layer Equations

Near a wall at high Reynolds number the flow splits into a thin viscous sheet and an inviscid outer stream. Inside the sheet, streamwise inertia is balanced by the impressed pressure gradient and by wall-normal viscous diffusion only; the cross-stream momentum equation degenerates to "pressure is constant across the layer," so the outer flow dictates \(p(x)\) while the layer supplies the no-slip correction. The layer thickness grows like \(Re^{-1/2}\), so it vanishes as \(Re\to\infty\) yet always exists.

Dispersion of Surface Gravity Waves

The squared frequency is the product of a gravitational restoring rate \(gk\) and a depth factor \(\tanh(kh)\) that runs smoothly from \(kh\) (shallow) to \(1\) (deep). Long waves feel the bottom (small \(kh\), factor \(\to kh\)); short waves do not (large \(kh\), factor \(\to 1\)). Because \(c\) depends on \(k\), the surface is a dispersive medium.

Kolmogorov's Four-Fifths Law

The mean cube of the longitudinal velocity increment is negative and grows linearly with separation. The negative sign is the signature of the direct energy cascade: on average larger eddies push energy toward smaller scales, biasing the increment distribution toward negative skewness. The magnitude divided by \(r\) is a direct, model-free measurement of the dissipation rate \(\varepsilon\). The full (unabridged) relation including the viscous term, \(S_3=-\tfrac45\varepsilon r+6\nu\,dS_2/dr\), holds at every scale.

Elasticity: Hooke's Law, Young's Modulus and Elastic Energy

The force is linear in displacement, the spring constant depends on the material and geometry, and the stored energy is quadratic in the extension.

Fluid Statics: Pressure, Depth and Archimedes' Principle

Pressure in a static fluid increases linearly with depth, and the buoyant force on any submerged object equals the weight of the fluid it displaces. The first result governs how pressure builds up in liquids; the second explains floating and sinking without reference to the object's material or shape.